Geodesic Convexity and Cartesian Products in Graphs

Geodesic Convexity and Cartesian Products in Graphs Tao Jiang Ignacio Pelayo Dan Pritikin September 23, 2004 Abstract In this work we investigate the behavior of various geodesic convexity parameters with respect to the Cartesian product operation for graphs. First, we show that the convex sets arising from geodesic convexity in a Cartesian product of graphs are exactly the same as the convex sets arising from the usual binary operation for making a convexity space out of the Cartesian product of any two convexity spaces. To be more precise, we prove that if (G 1, C 1 ), (G 2, C 2 ) are two graph (geodesic) convexity spaces and if (G 1 G 2, C) is the graph (geodesic) convexity space determined by the graph G 1 G 2, then C = C 1 C 2 = {A B A C 1, B C 2 }. Second, we study results involving a number of classical and graph-theoretic convexity parameters as applied to Cartesian products of graphs. For example, concerning geodetic numbers of graphs, we prove that for every two nontrivial graphs G, H such that gn(g) = p gn(h) = q 1, and that both bounds are tight. p gn(g H) pq q, 1 Introduction We consider only finite, simple, connected graphs. For undefined basic concepts we refer the reader to introductory graph theoretical literature, e.g., [9]. Given vertices u, v in a graph G we let d G (u, v) denote the distance between u and v in G. When there is no confusion, subscripts will be omitted. A u v path P is called a u v geodesic if it is a shortest u v path, that is, if E(P ) = d(u, v). The geodetically closed interval I[u, v] is the set of vertices of all u v geodesics. For S V, the geodetic closure I[S] of S is the union of all geodetically closed intervals I[u, v] over all pairs u, v S, i.e. I[S] = u,v S I[u, v]. A convexity on a finite set X is a family C of subsets of X (each such set called 0 Keywords: cartesian product, geodesic convexity, hull number, geodetic number, rank Dept. of Mathematics and Statistics, Miami University, Oxford, OH 45056, USA, jiangt@muohio.edu. Universitat Politècnica de Catalunya, ignacio.m.pelayo@upc.es Dept. of Mathematics and Statistics, Miami University, Oxford, OH 45056, USA, pritikd@muohio.edu 1

a convex set), which is closed under intersection and which contains both X and the empty set. The pair (X, C) is called a convexity space. A (finite) graph convexity space is a pair (G, C), formed by a finite connected graph G = (V, E) and a convexity C on V such that (V, C) is a convexity space satisfying that every member of C induces a connected subgraph of G. In this paper we consider only the so-called geodesic convexity, C g, defined as follows. A vertex set W V is called geodesically convex (or simply g-convex) if I[W ] = W. In a convexity space (X, C), the smallest convex set containing a given set A V is denoted [A] C and is called the convex hull of A. In the case where the space is a graph convexity space and the convexity used is geodesic convexity, we let [A] g denote the convex hull. A non-empty set A V is called a g-hull set if [A] g = V. In graph theory, the definition of G 1 G 2, the Cartesian product of graphs G 1 = (V 1, E 1 ) and G 2 = (V 2, E 2 ), is that it is the graph on vertex set V 1 V 2 in which vertices (x 1, x 2 ) and (y 1, y 2 ) are adjacent if and only if either x 1 = x 2 and y 1 y 2 E 2 or y 1 = y 2 and x 1 x 2 E 1. By contrast, in the theory of convexity spaces, a standard binary operation exists for producing what is called a convex product space, as follows. Given convexity spaces (X 1, C 1 ) and (X 2, C 2 ), the convex product space (X 1 X 2, C 1 C 2 ) on the set of points X 1 X 2 is defined by letting C 1 C 2 = {A B A C 1, B C 2 } [5]. In Section 2 of this paper we show that the operator as applied to the Cartesian product of two geodesic graph convexity spaces yields the same space as the geodesic graph convexity space induced by the usual Cartesian product graph formed from those graphs, connecting the closure of a set of vertices in the Cartesian product with the Cartesian product of the closures of the projections in each coordinate of the factor graphs. In the remainder of the paper we apply such basic connections so as to yield results concerning convexity parameters as applied to Cartesian products of graphs. Some such parameters are classical invariants (such as rank) in that they have been studied for some time as applied to arbitrary convexity spaces. Other such parameters are more recent ones (such as geodetic number) that are designed to apply specifically to graph convexity spaces. See [1] for some results of a similar nature. In Section 3 we give tight upper and lower bounds for the geodetic number of Cartesian products of graphs, in terms of the geodetic numbers of the factor graphs. In Section 4 2

we develop a formula for determining the rank of the Cartesian products of graphs in terms of the weak ranks of the factor graphs (which differ by at most one from the ranks of the factor graphs). 2 Geodesic Convexity and Cartesian Product In this section we reconcile whether the operator as applied to the Cartesian product of two geodesic graph convexity spaces yields the same space as the geodesic graph convexity space induced by the usual Cartesian product graph formed from those graphs. We begin with the following well known proposition. We omit its proof. Proposition 2.1 Let G, H be connected graphs. Given any two vertices (u, x), (v, y) in G H, we have d G H ((u, x), (v, y)) = d G (u, x) + d H (v, y). Furthermore, if P is a (u, x)-(v, y) geodesic in G H and P 1 and P 2 are the projections of V (P ) onto G and H, respectively, then P 1 induces a u v geodesic in G and P 2 induces a x y geodesic in H. For S a subset of a Cartesian product X 1 X 2 of any two sets X 1 and X 2, we make frequent use of the projection π i (S) of S onto coordinate i (i = 1, 2), given as usual by π 1 (S) = {u X 1 : v X 2, (u, v) S} and π 2 (S) = {v X 2 : u X 1, (u, v) S}. Often, when the context is clear, we write S 1 and S 2 as shorthand for π 1 (S) and π 2 (S) respectively. The following proposition tells us, when we create a convexity space by use of the operator, how to relate the convex hull of a set S of points in a convex product space to the projections of S on each coordinate. The proposition after that tells us the same information, but for when we create a graph convexity space on the graph Cartesian product of two graphs, and instead consider the geodesic convexity space associated with that resulting graph. Proposition 2.2 Let (X, C) = (X 1 X 2, C 1 C 2 ). If S X, then [S] C = [S 1 ] C1 [S 2 ] C2. Clearly the set [S 1 ] C1 [S 2 ] C2 specified is an element of C. A convex set containing S must be of the form A B with A C 1, B C 2, where S 1 A and S 2 B in order for S A B. Thus [S 1 ] C1 [S 2 ] C2 is the smallest element of C containing S, so it equals [S] C. 3

Proposition 2.3 Let G 1 = ((V 1, E 1 ), C 1 ) and G 2 = ((V 2, E 2 ), C 2 ), where the elements of C i are the g-convex sets in G i. Let G = (V, E) be the graph Cartesian product G 1 G 2 and let C be the set of g-convex sets in G. 1. If S V then, [S] C = [S 1 ] C1 [S 2 ] C2. 2. If a = (a 1, a 2 ), b = (b 1, b 2 ) V (G) then, I[a, b] = I[a 1, b 1 ] I[a 2, b 2 ]. 3. If S V then, I[S] I[S 1 ] I[S 2 ]. Consider any S C. We show that π 1 ([S] C ) C 1, as follows. Suppose for contradiction that for some x, x π 1 ([S] C ) that there exists an x x geodesic in G 1 containing a vertex z / π 1 ([S] C ). Then there exist (x, y), (x, y ) [S] C. The distance from (x, y) to (x, y ) is d G1 (x, x ) + d G2 (y, y ), so there exists an (x, y) (x, y ) geodesic that goes through (z, y) along the way to (x, y) before continuing on to (x, y ). Since [S] C C we have that (z, y) [S] C, so z π 1 ([S] C ), a contradiction. By symmetric argument, π 2 ([S] C ) C 2. Therefore [S i ] Ci π i ([S] C ) for i1, 2, since π i ([S] C ) C i and S i π i ([S] C ). Next we show that [S 1 ] C1 [S 2 ] C2 [S] C. Consider any (x, y) [S 1 ] C1 [S 2 ] C2. There exist x, y for which (x, y ) [S] C and (x, y) [S] C, since [S i ] Ci π i ([S] C ). There is an (x, y ) (x, y) geodesic that passes through (x, y), so (x, y) [S] C. Therefore [S 1 ] C1 [S 2 ] C2 [S] C. It is easy to see that S [S 1 ] C1 [S 2 ] C2 [S] C where [S 1 ] C1 [S 2 ] C2 C. Yet [S] C is the smallest element of C containing S, whereas [S] C is known to contain [S 1 ] C1 [S 2 ] C2, so [S] C = [S 1 ] C1 [S 2 ] C2, proving item 1. Concerning item 2, consider any x = (x 1, x 2 ) I[a, b], along an a b geodesic. Then since d(a, b) = d(a 1, b 1 ) + d(a 2, b 2 ), it must be for i = 1, 2 that x i is on an a i b i geodesic, so x I[a 1, b 1 ] I[a 2, b 2 ]. Conversely, consider any x = (x 1, x 2 ) I[a 1, b 1 ] I[a 2, b 2 ]. Then there exists an a 1 b 1 geodesic which follows a path P from a 1 to x 1 and from there on to b 1 via a path Q, and likewise an a 2 b 2 geodesic which follows a path R through x 2. Then the path from a = (a 1, a 2 ) to (x 1, a 2 ) (formed as in P while holding the second entry a 2 fixed) followed by the path from there to (x 1, x 2 ) and then on to x 1, b 2 (formed as in R while holding the first entry x 1 fixed) followed by the path from there to (b 1, b 2 ) (formed as in Q while holding the second entry b 1 fixed) yields an a b geodesic containing x. Therefore item 2 holds. 4

Concerning item 3, I[S] = a,b S I[a, b] = a,b S I[a 1, b 1 ] I[a 2, b 2 ] ( a 1,b 1 S I[a 1, b 1 ]) ( a 2,b 2 S I[a 2, b 2 ]) = I[S 1 ] I[S 2 ]. Now we can show, as promised, that the g-convex sets of the graphically defined G 1 G 2 turn out to be exactly those convex sets that we get by instead using the convexity operator in combining the g-convex sets for G 1 with the g-convex sets for G 2. Theorem 2.4 Let G 1 = ((V 1, E 1 ), C 1 ) and G 2 = ((V 2, E 2 ), C 2 ), where the elements of C i are the g-convex sets in G i. Let G = (V, E) be the Cartesian product graph G 1 G 2 and let C be the set of g-convex sets in G. Then 1. S C S i C i, i {1, 2}, and S = S 1 S 2 (where S i = π i (S)). 2. C = C 1 C 2. Part (1) of the previous proposition tells us that the g-convex subsets of V are precisely the Cartesian products A 1 A 2 where A i C i for i = 1, 2, since every g-convex set S must equal the Cartesian product of convex sets given for [S] C in the proposition and since the proposition shows that every such Cartesian product equals its closure. Therefore item 1 holds. Item 2 is simply the observation that C 1 C 2 produces the very same convex sets as does C. While fundamental, the result of Theorem 2.4 is not surprising. But neither should it be entirely obvious. For example, in R R, the most well known notion of convexity is that of Euclidean convexity, yet the operator applied to geodesic convexity spaces induced on the factor sets yields a very different convexity space, one in which the bounded nonempty convex sets are Cartesian products of bounded intervals of real numbers, i.e. horizontally and vertically aligned rectangles. Intuitively what is going on is that the standard distance metric in Cartesian products of graphs is the taxicab metric, whereas the familiar metric in R R is the Euclidean metric. 3 Hull Numbers and Geodetic Numbers of Cartesian Products Given a graph G, a subset S of V (G) is called a g-hull set for G if [S] g = V (G). The hull number of a graph G, denoted by hn(g), is the minimum cardinality of a g-hull set of V (G). We are in a position to give a simple proof of a generalization of the following known result. 5

Proposition 3.1 ([2]) If V (G) 2 then hn(g K 2 ) = hn(g). Theorem 3.2 hn(g H) = max{hn(g), hn(h)}. Let A = {a 1,, a p } be a minimum g-hull set for G and let B = {b 1,, b q } be a minimum g-hull set for H, where without loss of generality we can assume that p q. Consider any S V (G H). By Proposition 2.3, [S] g = [S 1 ] g [S 2 ] g, so S is a g-hull set for G H if and only if S 1 is a g-hull set for G and S 2 is a g-hull set for H. The choice S = {(a 1, b 1 ), (a 2, b 2 ),..., (a q, b q ), (a q+1, b q ), (a q+2, b q ),..., (a p, b q )} satisfies this requirement, with S = p, so hn(g) p. Also, hn(g H) p, since S must have at least hn(g) many elements for its projection S 1 to have at least hn(g) elements. Thus hn(g H) = p, completing the proof. Given a graph G, a subset S of V (G) is called a geodetic set for G if I[S] = V (G). The geodetic number of a graph G, denoted by gn(g), is the minimum cardinality of a geodetic set of V (G) [4]. Since I[S] need not be a convex set, whereas [S] g is necessarily convex, it turns out that the determination of the geodetic number of the Cartesian product of two graphs is more interesting than the determination of the hull number. Not surprisingly, the following lower bound result is related to the hull number for Cartesian products. The special case where H = K 2 has already appeared ([3]). Proposition 3.3 Let G and H be graphs. Then gn(g H) max{gn(g), gn(h)}, with equality when G, H are complete graphs. If S is a minimum geodetic set in G H then by item 3 of Proposition 2.3, V (G H) = I[S] I[S 1 ] I[S 2 ]. Therefore S 1, S 2 are geodetic sets in G, H respectively, so gn(g H) = S max{ S 1, S 2 } max{gn(g), gn(h)}, proving the lower bound. Consider complete graphs G, H with vertex sets V (G) = {u 1, u 2, u p } and V (H) = {v 1, v 2,, v q }, where without loss of generality p q. Then gn(g) = p and gn(h) = q. Let S = {(u 1, v 1 ), (u 2, v 2 ),, (u q, v q ), (u q+1, v q ), (u q+2, v q ),, (u p, v q )}. It is straightforward to verify that S is a geodetic set for G H. Hence, gn(g H) S p = max{gn(g), gn(h)} gn(g H), so equality holds. 6

Definition 3.4 Let G, H be graphs and G H the Cartesian product of G and H. Given x V (H), the subgraph of G H induced by {(u, x) u V (G)} is isomorphic to G. We call it the copy of G corresponding to x. Similarly, given u V (G), the subgraph of G H induced by {(u, x) x V (H)} is isomorphic to H, and we call it the copy of H corresponding to u. Given a path P and two vertices x, y on P, we use P [x, y] to denote the portion of P between x and y, inclusive. Theorem 3.5 Let G, H be graphs with gn(g) = p and gn(h) = q where p q 2. Then gn(g H) pq q. Let A = {a 1,, a p } be a geodetic set of G and B = {b 1,, b q } a geodetic set of H. Let S = A B {(a 1, b 1 ), (a 2, b 2 ),, (a q, b q )}. Then S = pq q. We show that S is a geodetic set of G H. Let (u, x) be any vertex in G H, where u V (G) and x V (H). We prove that (u, x) I G H (S). Since A is a geodetic set of G, there exist a i, a j A, where i j, such that u lies on a shortest a i, a j -path P in G. Similarly, since B is a geodetic set of H, there exist b k, b l B, where k l, such that x lies on a shortest b k, b l -path Q in H. We consider different cases. Case 1. i = j and k = l. In this case, u = a i and x = b k. If i k, then (u, x) S. Suppose then i = k. Let h [q] {i}; such h exists since q 2. Let L be a shortest path between (a h, b i ) and (a i, b i ) in the copy of G corresponding to b i and let L be a shortest path between (a i, b i ) and (a i, b h ) in the copy of H corresponding to a i. Then L L is a path in G H with length d G (a h, a i )+d H (b i, b h ) between (a h, b i ) and (a i, b h ). that contains (u, x). By Proposition 2.1, d G H ((a h, b i ), (a i, b h )) = d G (a h, a i )+d H (b i, b h ). Hence L L is a shortest path between (a h, b i ) and (a i, b h ) in G H. Furthermore, (a h, b i ), (a i, b h ) S since i h. Hence (u, x) I G H (S). Case 2. i < j, k = l. In this case, x = b k. Suppose first that k / {i, j}. Then (a i, b k ), (a j, b k ) S. Recall that P is a shortest a i, a j -path in G containing u. The copy of P in the copy of G corresponding to b k is a shortest path between (a i, b k ) and (a j, b k ) in G H that contains (u, x). Hence (u, x) I G H (S). 7

Suppose next that k = i. Let L be a copy of P in the copy of G corresponding to b i. Note that L contains (u, x). Let L be a shortest path between (a i, b i ) and (a i, b j ) in the copy of H corresponding to a i. By arguments similar to those in Case 1, L L is a shortest path between (a j, b i ) and (a h, b i ) in G H that contains (u, x). Since (a j, b i ), (a i, b i ) S, (u, x) I G H (S). The subcase where j = k can be handled in the same fashion. Case 3. i = j, k < l. This case can be handled in the same fashion as Case 2. Case 4. i < j, k < l. We consider three subcases. Subcase 4.1 i = k or j = l. Recall that P is a shortest path between a i and a j in G that contains u and Q is a shortest path between b k and b l in G that contains x. Let L 1 be a copy of P in the copy of G correspoding to b l. Let L 2 be a copy of Q in the copy of H corresponding to u. Let L 3 be a copy of P in the copy of G corresponding to x. Let L 4 be a copy of Q in the copy of H correpsonding to a j. It is straightforward to verify, using Proposition 2.1 that L 1 [(a i, b l ), (u, b l )] L 2 [(u, b l ), (u, x)] L 3 [(u, x), (a j, x)] L 4 [(a j, x), (a j, b k )] is a shortest path between (a i, b l ), (a j, b k ) in G H that contains (u, x). Since (a i, b l ), (a j, b k ) S, we have (u, x) I G H (S). Subcase 4.2 i k, j l. Let L 1 be a copy of P in the copy of G correspoding to b k. Let L 2 be a copy of Q in the copy of H corresponding to u. Let L 3 be a copy of P in the copy of G corresponding to x. Let L 4 be a copy of Q in the copy of H correpsonding to a l. It is straightforward to verify, using Proposition 2.1 that L 1 [(a i, b k ), (u, b k )] L 2 [(u, b k ), (u, x)] L 3 [(u, x), (a j, x)] L 4 [(a j, x), (a j, b l )] is a shortest path between (a i, b k ), (a j, b l ) in G H that contains (u, x). Since (a i, b k ), (a j, b l ) S, we have (u, x) I G H (S). Next, we show that the inequality in Theorem 3.5 is best possible. 8

Construction 3.6 Given positive integers p, t, let Dp t be a graph obtained as follows. Take p vertices x 1,, x p, take ( p 2) groups of vertices Wi,j for i, j [p], i < j, where the W i,j s are pairwise disjoint and also disjoint from {x 1,, x p } and each W i,j consists of t vertices. Next, for each pair i, j [p], i < j, join each of the t vertices in W i,j to both x i and x j. Finally, we add a new vertex z and join it to all the other vertices. The resulting graph is D t p. Lemma 3.7 Let p, t be positive integers such that t > p. Let G = D t p be defined as in Construction 3.6. Then gn(g) = p. We keep the notation used in Construction 3.6. First, note that G has diameter 2. Since S = {x 1,, x p } is clearly a geodetic set of G, gn(g) p. Let S be an arbitrary geodetic set of G. We need to prove that S p. If S contains all of x 1,, x p, then S p. So, we may assume that x l / S for some l [p]. let i [p] {l}. Without loss of generality suppose i < l. Let w be any vertex in W i,l. Suppose u / S, then since S is a geodetic set of G there exist u, v S such that w lies on a shortest u, v-path P in G. Such a path has length at least 2. Since G has diameter 2, P has length exactly 2. So u, v are two neighbors of w at distance two from each other in G. Since x i, x l, z are the only neighbors of w in G while z is at distance 1 from any other vertex, {u, v} = {x i, x l }. This contradicts our assumption that x l / S. Hence, S must contain all of W i,l. In particular, this implies that S W i,l = t > p. Proposition 3.8 Let p, q, t be positive integers such that t > pq q. Let G = D t p as defined in Construction 3.6. Let H = K q. Then gn(g) = p, gn(h) = q, and gn(g H) pq q. We keep the notation used in Construction 3.6. That gn(g) = p follows readily from Lemma 3.7, while gn(h) = q is trivial. Let S be a smallest geodetic set of G H. Suppose that S < pq q, we derive a contradiction. Suppose V (H) = V (K q ) = {v 1,, v q }. For each i [q], let G i denote the copy of G corresponding to v i. Since S < (p 1)q, by the pigeonhole principle, there exists some l [q] such that S V (G l ) < p 1. For each vertex u V (G), we use u (l) to denote its image in G l. Recall the notation used in Construction 3.6. Since S V (G l ) p 2, there exist i, j [p] such that x (l) i, x (l) j / S. Consider the set W i,j. For each w W i,j, let A w = {w (1),, w (q) }, i.e. A w is the set 9

of the q images of w in G H. We show that S A w for each w W i,j. If w (l) S, then there is nothing to prove. So, we may assume that w (l) / S. Then there exist vertices a, b S such that w (l) lies on a shortest a, b-path P in G H. Since G has diameter 2 by Lemma 3.7 and H = K q, G H has diameter 3. Hence P has length at most 3. In particular, this implies that either a or b is a neighbor of w (l) in G H. Without loss of generality, suppose a is a neighbor of w (l) in G H. Note that the closed neighborhood of w (l) in G H is A w {x (l) i x (l) i, x (l) j, x (l), z(l) } and a, b / {x (l), x (l) } since / S. If a z (l) then a A w and thus S A w. So we may assume that a = z (l). Since z is a dominating vertex in G, a = z (l) is at distance at most 2 from any other vertex in G H. In particular, this implies that P has length 2, in which case b is also a neighbor of w (l). Hence, b A w. Therefore S A w. We have shown that for each w W i,j, S A w. Since W i,j contains t different w s, and the A ws are clearly pairwise disjoint for different w s, we have S t > pq q, contradicting S < pq q. The contradiction completes the proof. j i j We summarize Proposition 3.3, Theorem 3.5, and Proposition 3.8 as follows. Proposition 3.9 Let G and H be nontrivial graphs. We have max{gn(g), gn(h)} gn(g H) gn(g) gn(h) min{gn(g), gn(h)}. Furthermore, both inequalities above are best possible. 4 Classical Invariants and Cartesian Products One classical convexity invariant studied in the literature as applied to arbitrary convexity spaces is that of the rank of a convexity space [8], a notion aptly named because of its similarities to the rank of a matrix or matroid. We present a newly defined variation, called the weak rank, and show how it applies to the determination of the rank of a graph. First we introduce some definitions concerning dependent and independent sets, as one might expect. Consider a convexity space (X,C) and a subset S of X and an element s of S. We say that s is (convexly) dependent in S if s [S s] C, and call s (convexly) independent in S otherwise. S is called convexly dependent if x [S x] C for at least one x S (i.e. some element is dependent in S) and 10

called convexly independent otherwise. S is called strongly convexly dependent if x [S x] C for at least two elements x S, and called weakly convexly independent otherwise, i.e., when at most one exceptional such element x exists in S. We use the abbreviations sc-dependent and wc-independent for the terms strongly convexly independent and weakly convexly independent, respectively. When wc-independent S is such that a particular element s S is dependent in S we say that s is the sole dependent in S. Observe that (s is dependent in S) s [S s] C [S] C = [S s] C. Also observe that convex independence is a hereditary property, as is wc-independence. That is, every subset of a convexly independent set is convexly independent, and every subset of a wc-independent set is wc-independent. Lastly observe that if s is the sole dependent in S then S s is convexly independent, since for each x S s it is the case that [(S s) x] C [S x] C, so x / [(S s) x] C since x / [S x] C. Definition 4.1 The rank of a graph G is the maximum cardinality of a convexly independent set, and is denoted by r = r(g). The weak rank of a graph G is the maximum cardinality of a wcindependent set, and is denoted by r 1 = r 1 (G). Proposition 4.2 r r 1 r + 1. Certainly, r r 1. For proving the upper bound, suppose for contradiction that a wcindependent set S of cardinality exceeding r + 1 exists. By maximality of r, the set S must be convexly dependent, so let s be the sole dependent in S. Then S s is convexly independent and yet has cardinality exceeding r, a contradiction, completing the proof. We introduced the concept of the weak rank of a set because it allows us in this section to make a connection between the rank of a Cartesian product of graphs and the weak ranks of the factor graphs. Meanwhile, we still develop one more claim toward that goal. Proposition 4.3 Consider any connected graphs G and H and any convexly independent subset S of V (G H). Then each (a, b) S must be of one or both of the following types. Type 1: a / π 1 (S (a, b)) and a / [S 1 a]. Type 2: b / π 2 (S (a, b)) and b / [S 2 a]. Consider any (a, b) S. Suppose for contradiction that there exist elements (a, b ) and (a, b) of [S (a, b)]. Then there would exist an (a, b ) (a, b) geodesic in G H that passes through 11

(a, b), contradicting that (a, b) / [S (a, b)]. Therefore either no such b exists or no such a exists. Suppose first that no such b exists, i.e. a / π 1 ([S (a, b)]). By Proposition 2.3, a / [π 1 (S (a, b))], from which (a, b) is of Type 1. By symmetry, the remaining case (in which no such a exists) requires that (a, b) is of Type 2. Finally, we can now state and prove the connection between the rank of a Cartesian product of graphs and the weak ranks of the factor graphs. Theorem 4.4 For any nontrivial connected graphs G 1 and G 2, r(g 1 G 2 ) = r 1 (G 1 ) + r 1 (G 2 ) 2. For i = 1, 2, let S i be a largest wc-independent set of vertices in G i, and select a particular element x i of S i, being sure to select the sole dependent in S i if S i happens to fail to be convexly independent. Then let S = {(a, x 2 ) a S 1 x 1 } {(x 1, b) b S 2 x 2 }. Clearly S = r 1 (G 1 ) + r 1 (G 2 ) 2. We show that S is convexly independent, establishing that r(g 1 G 2 ) r 1 (G 1 ) + r 1 (G 2 ) 2. To show that S is convexly independent, it suffices to show for each a S 1 x 1 that (a, x 2 ) / [S (a, x 2 )], i.e. that each (a, x 2 ) is independent in S, since by symmetry it follows that each (x 1, b) is independent in S as well, so that each element of S is independent in S. Now, by Proposition 2.3, π 1 ([S (a, x 2 )]) equals [π 1 (S (a, x 2 ))]. Since (a, x 2 ) is the only vertex of S having a in the first coordinate, this latter set in turn equals [S a]. Since S 1 is wc-independent and a is not its sole dependent, we have that a / [S a] = π 1 ([S (a, x 2 )]), proving that (a, x 2 ) / [S (a, x 2 )], completing our proof that r(g 1 G 2 ) r 1 (G 1 ) + r 1 (G 2 ) 2. Next we show that r(g 1 G 2 ) r 1 (G 1 ) + r 1 (G 2 ) 2. Let T 1 = {1st coordinates of the Type 1 vertices of S} and let T 2 = {2nd coordinates of vertices of S that are of Type 2 but not Type 1}, so T 1 + T 2 = r(g 1 G 2 ). First suppose that S 1 T 1 = 0. Then T 2 = since no element of T 1 = S 1 can appear more than once among the first coordinates of vertices of S. Then S 1 is convexly independent, giving us that 12

r(g 1 G 2 ) r(g 1 ) + 0 r 1 (G 1 ) + r 1 (G 2 ) 2, since any 2-subset of V (G 2 ) forms a wc-independent set. Similarly, r(g 1 G 2 ) r 1 (G 1 ) + r 1 (G 2 ) 2 if S 2 T 2 = 0. So, we can assume that there exist vertices x 1 S 1 T 1 and x 2 S 2 T 2. Now, T 1 {x 1 } is wc-independent, since each first coordinate of a Type 1 vertex is independent in S 1 (by the meaning of Type 1 ), so is also independent in T 1 {x 1 }. Therefore T 1 r 1 (G 1 ) 1. Similarly, T 2 r 1 (G 2 ) 1, from which r(g 1 G 2 ) = T 1 + T 2 r 1 (G 1 ) + r 1 (G 2 ) 2, completing the proof. We recast the result of Theorem 4.4 in the context of related results concerning other classical convexity invariants. S is called redundant if [S] C = [S a] C. Observe that every convexly dependent set is a S redundant, since for every a A, a [A a] C [A] C = [A a] C. Irredundancy is not an hereditary property. A subset S of a convexity space (X, C) is said to be weakly irredundant if it is not strongly redundant, i.e., if there exists some element x in S satisfying the following condition: [S a] C [S] C. a S x Certainly, every irredundant set is weakly irredundant, since [S] C [S a] C = [S x] C ( [S a] C ). a S a S x The Caratheodory number c = c(g) is the maximum cardinality of an irredundant set. Observe that c r. To show a connection between the result of Theorem 4.4 and a result of Sierksma [5] concerning the Caratheodory number, we introduce the notion of the weak irredundancy number c 1 of a convexity space (V, C) as the maximum cardinality of a weakly irredundant set. Proposition 4.5 1. Every weakly irredundant set is weakly convexly independent. 2. c 1 r 1. Let S be a non-weakly convexly independent set. Let y, z S, y z, s.t. [S y] C = [S z] C = [S] C. Hence, for every x S, [S] C = a S x [S a] C. In consequence, S is not a weak 13

redundant set, proving item 1. Item 2 is an immediate consequence. Proposition 4.6 c c 1 c + 1. Certainly, c c 1. For proving the upperbound, suppose S is a set of cardinality c + 2, this means that S x is redundant, for every x S. Hence, [S] C = [S a] C = [S x] C ( [S a] C ) = ( [S x a] C ) ( [S a] C ). a S a S x a S x a S x In consequence, as [S x a] C [S a] C for every x S, we derive that S is not a weakly irredundant set. The previous result is much like Proposition 4.2 concerning weak rank. The next result is the weak irredundancy number counterpart of Theorem 4.4. We had proved this next result independently, only to realize that it is equivalent to the theorem of Sierksma [5] listed immediately after. Consequently, we omit the proof, but call attention to the obvious connections between these results concerning rank and Caratheodory/irredundancy numbers. Theorem 4.7 c(g 1 G 2 ) = c 1 (G 1 ) + c 1 (G 2 ) 2. Theorem 4.8 ([5]) c(g 1 G 2 ) = c(g 1 ) + c(g 2 ) k, where k {0, 1, 2} is the number of factors for which e c. A set of vertices S is called exchange-dependent, or just e-dependent, if [S x] C a S x [S a] C for all x S, and exchange independent otherwise. The exchange number, e = e(g), is the maximum cardinality of an exchange independent set. In [6], G. Sierksma proved that e 1 c. Notice that a set is strongly redundant if and only if it is both redundant and exchange dependent, and hence we can assure that c 1 = max{e, c}. Theorem 4.9 ([7]) e(g 1 G 2 ) = max{e(g 1 ), c(g 1 )} + max{e(g 2 ), c(g 2 )} 1 Corollary 4.10 e(g 1 G 2 ) = c 1 (G 1 ) + c 1 (G 2 ) 1 = c(g 1 G 2 ) + 1 14

5 Concluding Remarks While we have defined the rank, Caratheodory number and exchange number only in the context of applying them to graphs, all three are traditionally understood (using the same definitions given here) as applying to arbitrary convexity spaces [8]. Results given here concerning Cartesian products of two graphs G 1, G 2 generalize easily to Cartesian products of arbitrarily many graphs. For example, more generally than we bothered to state in Proposition 3.3, it is true for all graphs G 1,..., G k that gn(g 1 G 2... G k ) max{gn(g i )}. Here are two more miscellaneous known results concerning convexity parameters applied directly to Cartesian products of graphs. Theorem 5.1 ([5]) Let (X, C) = (X 1 X 2, C 1 C 2 ). Then h = max{h 1, h 2 }, where h i = h(x i ), where h denotes the Helly number of the corresponding convexity space. Theorem 5.2 ([1]) con(g 1 G 2 ) = max{ V 2 con(g 1 ), V 1 con(g 2 )}, where con(g), called the convexity number, is the maximum cardinality of a proper g-convex set of V (G). Whereas e 1 c, we know of no example of a graph G for which c(g) > e(g). We leave it as an open problem as to whether e 1 c e holds in general for all graphs. References [1] S.R. Canoy and I.J.L. Garces, Convex sets under some graph operations, Graphs and Combinatorics, 18 (4) (2002), 787-793. [2] G. Chartrand, F. Harary and P. Zhang, On the Hull Number of a Graph, Ars Combinatoria 57 (2000), 129-138. [3] G. Chartrand, F. Harary and P. Zhang, On the Geodetic Number of a Graph, Networks 39 (1) (2002), 1-6. [4] F. Harary, E. Loukakis and C. Tsouros, The geodetic number of a graph, Math. Comput. Modelling 17 (11) (1993), 89-95. 15

[5] G. Sierksma, Caratheodory and Helly-numbers of convex product structures, Pacific J. Math., 61 (1975) 275-282. [6] G. Sierksma, Relationships between Carathéodory, Helly, Radon and exchange numbers of ocnvexity spaces, Nieuw Archief voor Wisk., 25 (2) (1977) 115-132. [7] V. P. Soltan, Substitution numbers and Carathéodory numbers of the Cartesian product of convexity structures, Ukrain. Geom. Sb., 24 (1981) 104-108. [8] M. Van de Vel, Theory of convex structures, North-Holland, Amsterdam, MA (1993). [9] D. B. West, Introduction to Graph Theory, Prentice-Hall, New Jersey, (1999). 16